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Mathematics · University

Gradients, tangent planes and extrema

The gradient collects first partial derivatives and determines directional derivatives through a dot product.

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Prerequisites

Partial derivatives and vector dot products.

Study notes

Gradient

The gradient collects first partial derivatives and determines directional derivatives through a dot product.

Tangent plane

Linearization at a differentiable point uses f(a,b)+fₓ(a,b)(x−a)+fᵧ(a,b)(y−b).

Stationary points

A zero gradient is necessary for an interior differentiable extremum, not sufficient.

Hessian test

In two variables, positive determinant and positive fₓₓ identify a strict local minimum; negative determinant identifies a saddle.

Worked examples

For f(x,y)=x²+3y², find fₓ at (2,1).

Differentiate with respect to x to obtain 2x, then substitute x=2.

4

Write the tangent plane to z=x²+y² at (1,2,5).

Credit the correct base height and both gradient coefficients.

z=5+2(x−1)+4(y−2), equivalently z=2x+4y−5.

Check your understanding

  1. For f(x,y)=x²+3y², find the directional derivative at (1,1) in direction (3/5,4/5).

  2. What kind of stationary point is the origin for f=x²−y²?

    1. A strict local maximum
    2. A point with no partial derivatives
    3. A saddle point
    4. A strict local minimum

  3. Find the determinant of the Hessian of f=2x²+3xy+5y².

  4. Why should the direction vector be normalized before calculating a directional derivative?

Answers and marking guidance

Answers and marking notes
  1. 6

    The gradient is (2,6). Its dot product with (3/5,4/5) is 6/5+24/5=6.

  2. A saddle point

    The function increases along the x-axis and decreases along the y-axis.

  3. 31

    The Hessian is [[4,3],[3,10]]; its determinant is 40−9=31.

  4. A directional derivative is conventionally the rate per unit distance in that direction. Without normalization, the dot product is scaled by the vector length.

    Credit the unit-distance interpretation and the scaling effect.

Further reading and review references

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